We define non-pluripolar products of an arbitrary number of closed positive $(1,1)$-currents on a compact Kähler manifold $X$. Given a big $(1,1)$-cohomology class $\a$ on $X$ (i.e.~a class that can be represented by a strictly positive current) and a positive measure $\mu$ on $X$ of total mass equal to the volume of $\a$ and putting no mass on pluripolar subsets, we show that $\mu$ can be written in a unique way as the top degree self-intersection in the non-pluripolar sense of a closed positive current in $\a$. We then extend Kolodziedj's approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if $\mu$ has $L^{1+\e}$-density with respect to Lebesgue measure. If $\mu$ is smooth and positive everywhere, we prove that $T$ is smooth on the ample locus of $\a$ provided $\a$ is nef. Using a fixed point theorem we finally explain how to construct singular Kähler-Einstein volume forms with minimal singularities on varieties of general type.